Magnetic Resonance in Quantum - PDF Free Download

Magnetic Resonance in Quantum Information Christian Degen Spin Physics and Imaging group Laboratory for Solid State Physics www.spin.ethz.ch

Content Features of (nuclear) magnetic resonance Brief History of NMR Example 1: Liquid-state NMR (=> Factoring of 15) Example 2: Example 2: Quantum information using single spins in diamond

Magnetic nuclei Blue => Nuclei with spin I = ½ Red => Nuclei with spin I > ½ Grey => Nuclei with spin zero, or not known

Time and Energy scales in NMR & EPR Nuclei (Protons) Electrons Relevant energy: Zeeman energy 43 MHz/T 28 GHz/T in magnetic field (up to ~24Tesla) (~0.2µeV/T) (~120µeV/T) Polarization (room temperature) ~10-5 ~1% Spin relaxation time T 1 0.1 10 5 s 10-9 1 s Spin coherence time T 2 ~1 100µs (solids) 10-9 10-3 s ~10ms 10s (liquids) Spin rotations (pulses) ~1-10µs ~1-100 ns Swaps (spin-spin couplings) ~10µs 10ms ~0.1-10µs

NMR largely satisfies the DiVincenzo criteria Qubits: nuclear spins ½ in B0 field ( and as 0 and 1) Quantum gates: RF pulses and delay times ( ) Input: Boltzmann distribution (room temperature) Readout: detect spin states with RF coil Coherence times: easily several seconds (Gershenfeld & Chuang 1997, Cory, Havel & Fahmi 1997) 4

Brief history of NMR Literature suggestion: A. Abragam, Time reversal: An autobiography (Oxford University Press, 1989) 1922 Stern-Gerlach experiment 1939 Rabi s molecular beam experiment [Nobel Prize 1944] 1945-1950 Bloch equations [Nobel Prize 1952] 1950-1965 Physical basis of NMR: Relaxation, Couplings, DNP, etc. ~1965 Fourier-transform NMR (Ernst) [Nobel Prize 1991] 1973 Invention of magnetic resonance imaging [Nobel Prize 2003] ~1975 Multi-dimensional NMR 1975-1990 1990 Advanced multi-pulse techniques (decoupling, ) 1980- Application in Structural Biology [Nobel Prize 2002] 1995 - Application in Quantum engineering g

Example 1: Liquid-state id t NMR Principles of NMR QC Techniques for qubit control State of the art & Outlook => NMR Quantum Computing Slides courtesy of Lieven Vandersypen Then: IBM Almaden, Stanford University Now: Kavli Institute of NanoScience, TU Delft

How to factor 15 with NMR? 2

Liquid-state NMR Survey of NMR quantum computing Principles of NMR QC Techniques for qubit control Summary: Pros & Cons

Nuclear spin Hamiltonian Single spin ½ 5

Nuclear spin Hamiltonian Multiple spins without qubit/qubit coupling chemical shifts of the five F qubits 1H 13C 126 15NN -51 19F 470 31P 202 MHz 500 ~25mK (at 11.7 Tesla) qubit level l separation 6

Hamiltonian with RF field single-qubit rotations σz σx σy rotating wave approximation typical strength Ix, Iy : up to 100 khz Rotating frame Lab frame 7

Nuclear spin Hamiltonian Coupled spins J>0: antiferro mag. coupling term J<0: ferro-mag. Typical values: J up to few 100 Hz 16 configurations solid (dashed) d) lines are (un)coupled levelsl 8

Making room temperature t spins look cold (Cool to mk) Optical pumping DNP, warm Effective pure state (Gershenfeld&Chuang, Science 97, Cory, Havel & Fahmi, PNAS 97) equalize population quasi cold cold warm Look exactly like cold spins! quasi cold [Hz] [Hz] 10

Effective pure state preparation (1) Add up 2 N -1 experiments (Knill,Chuang,Laflamme,, PRA 1998) + + = Later : (2 N - 1) / N experiments (Vandersypen et al., PRL 2000) prepare equal population (on average) and look at deviations from equilibrium. (2) Work in subspace (Gershenfeld&Chuang, Science 1997) compute with qubit states that have the same energy and thus the same population. 11

Read-out in NMR Phase sensitive detection 0 x y positive signal for 0> (in phase) 1 x z Y90 y x x z y y negative signal for 1> (out of phase) 12

NMR Spectrometer Transmission / Receiver coil Computer Console Superconducting magnet

Liquid-state NMR Survey of NMR quantum computing Principles of NMR QC Techniques for qubit control Summary: Pros & Cons

Off-resonance pulses and spin-selectivity spectral content of a square pulse off-resonant pulses induce eff. σz rotation in addition to σx,y may induce transitions in other qubits 17

Pulse shaping for improved spin-selectivity gaussian pulse profile gaussian spectrum less cross-talk 18

Missing i coupling terms: Swap How to couple distant qubits with only nearest neighbor physical couplings? Missing couplings: swap states along qubit network SWAP12 = CNOT12 CNOT21 CNOT12 1 2 3 4 only a linear overhead... 19

Undesired couplings: refocus remove effect of coupling during delay times opt. 1: act on qubit B X B 180 X B 180 opt. 2: act on qubit A There exist efficient extensions for arbitrary coupling networks Refocusing can also be used to remove unwanted Zeeman terms 20

Composite pulses Example: Y90X180Y90X Y corrects for under/over-rotation z corrects for off-resonance z x y x y However: doesn t work for arbitrary input state But: there exist composite pulses that work for all input states 22

Molecule l selection A quantum computer is a known molecule. Its desired d properties are: spins 1/2 ( 1 H, 13 C, 19 F, 15 N,...) long T 1 s and T 2 s heteronuclear, or large chemical shifts good dj-coupling network (clock-speed) stable, available, soluble,... required to make spins of same type addressable 24

Quantum computer molecules l (1)

Quantum computer molecules l (2)

Liquid-state NMR Survey of NMR quantum computing Principles of NMR QC Techniques for qubit control Summary: Pros & Cons

The good news Quantum computations have been demonstrated in the lab A high degree of control was reached, permitting hundreds of operations in sequence A variety of tools were developed for accurate unitary control over multiple coupled qubits useful in other quantum computer realizations Spins are natural, attractive qubits 29

The main issue: Scaling We do not know how to scale liquid NMR QC Main obstacles: Signal after initialization ~ 1 / 2 n [at least in practice] Coherence time typically goes down with molecule size We have not yet reached the accuracy threshold... 30

Main sources of errors in NMR QC Early on (heteronuclear molecules) inhomogeneity RF field Later (homonuclear molecules) l Finally J coupling during RF pulses decoherence 31

Electron spin qubits SL SR Kane, Nature 1998 Loss & DiVincenzo, PRA 1998 Gruber, Science, 1997

Quantum information using single spins in diamond Literature hint: F. Jelezko, J. Wrachtrup, Single defect centres in diamond: A review Phys. stat. sol. (a) 203, 3207 (2006).

Single spins in diamond Diamond nitrogen vacancy center (NV Center) - Principles of Optically-detected magnetic resonance (ODMR) - Initialization, manipulation, read-out of NV electron spin Environment: Diamond host material Applications

Optically-detected magnetic resonance (ODMR) Idea: Detect t optical instead of microwave/radio-frequency photons Requirement: Optical photon must be correlated with spin state σ = or Intensity I Wavelength λ Polarization P Intensity I (σ) I(σ) Wavelength λ (σ) Polarization P (σ)

Optically detected magnetic resonance Some History Optical detection of paramagnetism in phenantren (Kwiram, 1967) ( 1980ies: Invention of single molecule spectroscopy ) Optical detection of magnetic resonance in a single pentacene molecule Wrachrup, Moerner (1993) Optical detection of single nitrogen vacancy centers in diamond: Gruber, Wrachtrup (1997)

Single spin detection of NV centers Gruber, Science Si 1997

Readout and Polarization of the NV center Conduction band 5.5 ev 3 E ±1 0 532 nm 1.945 ev 0 630 800 nm 1 A Readout: m s = 0 scatters 30% more light than m s = ±1 Many readout cycles needed to detect spin state 3 A ±1 0 10 20 ns >100 ns Polarization ( Reset ): Illumination pumps spin into m s = 0 state (>90% fidelity) 29GHz 2.9 Valence band

Spin Hamiltonian Zero field splitting Electron Zeeman Hyperfine coupling to 14 N (2.87 GHz) (2.8 MHz/Gauss) (~2.2 (2.2 MHz) Jelezko, Journal of Phys. Cond Mat. 2006

Spin Hamiltonian Zeeman Hyperfine

NV centers in diamond Some properties: p Single photon emitter Optically stable: No bleaching, no blinking High quantum yield ~70% Room temperature Long spin coherence times >1ms In nanodiamonds down to 5 nm Excellent chemical stability Non-toxic biomarkers Inexpensive! NitrogenVacancy defect in diamond

Fabrication Ion implantation NV centers implanted into single crystal diamond, then patterned with waveguide Toyli, Nano Letters, 2010

Rapid spin manipulation by waveguides Rabi Oscillations Spin inversion within ~ 1 ns Fuchs, Science, 2009

Spin environment Nuclear spins Electron spins 14 NV nuclear spin (I=1) Nearby N defect electron spins 13 C spins (I=½) (S=1/2) ( P 1 or C centers ) 1.03% natural abundance Nearby other NV center (S=1) First shell: 3x dangling bonds Second, third, shell Surface spins (electronic, nuclear) Distant: Classical spin bath Surface spins (adsorbates)

Other electronic spins Nearby N defect with S = ½ ( P1 center ) Gaebel, Nature Physics, 2006

Coupling between two NV centers Neumann, Nature Physics, 2010

The 13 C nuclear spin environment Near 13 C spins resolved lines Intermediate 13 C spins lines not resolved, but still moderate hyperfine coupling Distant 13 C spins very weak hyperfine coupling ( classical ensemble )

Specific nearby 13 C Two qubits: NV + 13 C Three qubits: NV + 13 C+ 13 C Dutt, Science, 2007

Repetitive readout Store and Readout NV spin state in nearby 13 C nucleus init calculation store readout n Example: Rabi oscillation Jiang, Science, 2009

Distant 13 C spin bath Periodic revivals of spin echo with 13 C Larmor frequency Childress, Science, 2006

Eliminating 13 C nuclear spin bath The materials scientists way 13 C content Spin coherence time T 2 20.7% 0.010 ms 8.4 % 0.12 ms 1.1 % (natural) 0.65 ms 0.4% 1.8 ms Balasubramanian, Nature Materials, 2009

Eliminating 13 C nuclear spin bath Multi pulse decoupling (l (also dynamical decoupling ) the spectroscopists way Carr Purcell Meiboum Gibbs sequence

Hybrid implementations NV centers to superconducting microwave resonator Arrangement Transmission vs. Frequency Kubo, PRL, 2010

Semba, Nature, 2011 Flux qubit Vacuum Rabi splitting Arrangement

Arcizet, Nature Physics, 2011 Nanomechanical resonators

Application in nanoscale magnetic imaging Measure nanoscale magnetic fields from: Single electron spins Nuclear spins in (bio)molecules Superconductors Magnetic nanostructures Quantized currents in mesoscopics Degen, Appl. Phys. Lett, 2008 Disk drive (Rondin, arxiv, 2011)